- FindStat

Identifier

Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000508: Standard tableaux ⟶ ℤ

Values

[1,2] => [[1,2]] => 4

[2,1] => [[1],[2]] => 0

[1,2,3] => [[1,2,3]] => 9

[1,3,2] => [[1,2],[3]] => 4

[2,1,3] => [[1,3],[2]] => 0

[2,3,1] => [[1,2],[3]] => 4

[3,1,2] => [[1,3],[2]] => 0

[3,2,1] => [[1],[2],[3]] => 1

[1,2,3,4] => [[1,2,3,4]] => 16

[1,2,4,3] => [[1,2,3],[4]] => 10

[1,3,2,4] => [[1,2,4],[3]] => 6

[1,3,4,2] => [[1,2,3],[4]] => 10

[1,4,2,3] => [[1,2,4],[3]] => 6

[1,4,3,2] => [[1,2],[3],[4]] => 6

[2,1,3,4] => [[1,3,4],[2]] => 0

[2,1,4,3] => [[1,3],[2,4]] => 0

[2,3,1,4] => [[1,2,4],[3]] => 6

[2,3,4,1] => [[1,2,3],[4]] => 10

[2,4,1,3] => [[1,2],[3,4]] => 4

[2,4,3,1] => [[1,2],[3],[4]] => 6

[3,1,2,4] => [[1,3,4],[2]] => 0

[3,1,4,2] => [[1,3],[2,4]] => 0

[3,2,1,4] => [[1,4],[2],[3]] => 2

[3,2,4,1] => [[1,3],[2],[4]] => 0

[3,4,1,2] => [[1,2],[3,4]] => 4

[3,4,2,1] => [[1,2],[3],[4]] => 6

[4,1,2,3] => [[1,3,4],[2]] => 0

[4,1,3,2] => [[1,3],[2],[4]] => 0

[4,2,1,3] => [[1,4],[2],[3]] => 2

[4,2,3,1] => [[1,3],[2],[4]] => 0

[4,3,1,2] => [[1,4],[2],[3]] => 2

[4,3,2,1] => [[1],[2],[3],[4]] => 0

[1,2,3,4,5] => [[1,2,3,4,5]] => 25

[1,2,3,5,4] => [[1,2,3,4],[5]] => 18

[1,2,4,3,5] => [[1,2,3,5],[4]] => 14

[1,2,4,5,3] => [[1,2,3,4],[5]] => 18

[1,2,5,3,4] => [[1,2,3,5],[4]] => 14

[1,2,5,4,3] => [[1,2,3],[4],[5]] => 13

[1,3,2,4,5] => [[1,2,4,5],[3]] => 8

[1,3,2,5,4] => [[1,2,4],[3,5]] => 7

[1,3,4,2,5] => [[1,2,3,5],[4]] => 14

[1,3,4,5,2] => [[1,2,3,4],[5]] => 18

[1,3,5,2,4] => [[1,2,3],[4,5]] => 11

[1,3,5,4,2] => [[1,2,3],[4],[5]] => 13

[1,4,2,3,5] => [[1,2,4,5],[3]] => 8

[1,4,2,5,3] => [[1,2,4],[3,5]] => 7

[1,4,3,2,5] => [[1,2,5],[3],[4]] => 9

[1,4,3,5,2] => [[1,2,4],[3],[5]] => 7

[1,4,5,2,3] => [[1,2,3],[4,5]] => 11

[1,4,5,3,2] => [[1,2,3],[4],[5]] => 13

[1,5,2,3,4] => [[1,2,4,5],[3]] => 8

[1,5,2,4,3] => [[1,2,4],[3],[5]] => 7

[1,5,3,2,4] => [[1,2,5],[3],[4]] => 9

[1,5,3,4,2] => [[1,2,4],[3],[5]] => 7

[1,5,4,2,3] => [[1,2,5],[3],[4]] => 9

[1,5,4,3,2] => [[1,2],[3],[4],[5]] => 6

[2,1,3,4,5] => [[1,3,4,5],[2]] => 0

[2,1,3,5,4] => [[1,3,4],[2,5]] => 0

[2,1,4,3,5] => [[1,3,5],[2,4]] => 0

[2,1,4,5,3] => [[1,3,4],[2,5]] => 0

[2,1,5,3,4] => [[1,3,5],[2,4]] => 0

[2,1,5,4,3] => [[1,3],[2,4],[5]] => 0

[2,3,1,4,5] => [[1,2,4,5],[3]] => 8

[2,3,1,5,4] => [[1,2,4],[3,5]] => 7

[2,3,4,1,5] => [[1,2,3,5],[4]] => 14

[2,3,4,5,1] => [[1,2,3,4],[5]] => 18

[2,3,5,1,4] => [[1,2,3],[4,5]] => 11

[2,3,5,4,1] => [[1,2,3],[4],[5]] => 13

[2,4,1,3,5] => [[1,2,5],[3,4]] => 5

[2,4,1,5,3] => [[1,2,4],[3,5]] => 7

[2,4,3,1,5] => [[1,2,5],[3],[4]] => 9

[2,4,3,5,1] => [[1,2,4],[3],[5]] => 7

[2,4,5,1,3] => [[1,2,3],[4,5]] => 11

[2,4,5,3,1] => [[1,2,3],[4],[5]] => 13

[2,5,1,3,4] => [[1,2,5],[3,4]] => 5

[2,5,1,4,3] => [[1,2],[3,4],[5]] => 5

[2,5,3,1,4] => [[1,2,5],[3],[4]] => 9

[2,5,3,4,1] => [[1,2,4],[3],[5]] => 7

[2,5,4,1,3] => [[1,2],[3,5],[4]] => 7

[2,5,4,3,1] => [[1,2],[3],[4],[5]] => 6

[3,1,2,4,5] => [[1,3,4,5],[2]] => 0

[3,1,2,5,4] => [[1,3,4],[2,5]] => 0

[3,1,4,2,5] => [[1,3,5],[2,4]] => 0

[3,1,4,5,2] => [[1,3,4],[2,5]] => 0

[3,1,5,2,4] => [[1,3,5],[2,4]] => 0

[3,1,5,4,2] => [[1,3],[2,4],[5]] => 0

[3,2,1,4,5] => [[1,4,5],[2],[3]] => 3

[3,2,1,5,4] => [[1,4],[2,5],[3]] => 3

[3,2,4,1,5] => [[1,3,5],[2],[4]] => 0

[3,2,4,5,1] => [[1,3,4],[2],[5]] => 0

[3,2,5,1,4] => [[1,3],[2,5],[4]] => 0

[3,2,5,4,1] => [[1,3],[2,4],[5]] => 0

[3,4,1,2,5] => [[1,2,5],[3,4]] => 5

[3,4,1,5,2] => [[1,2,4],[3,5]] => 7

[3,4,2,1,5] => [[1,2,5],[3],[4]] => 9

[3,4,2,5,1] => [[1,2,4],[3],[5]] => 7

[3,4,5,1,2] => [[1,2,3],[4,5]] => 11

[3,4,5,2,1] => [[1,2,3],[4],[5]] => 13

[3,5,1,2,4] => [[1,2,5],[3,4]] => 5

[3,5,1,4,2] => [[1,2],[3,4],[5]] => 5

[3,5,2,1,4] => [[1,2],[3,5],[4]] => 7

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$F_{2} = 1 + q^{4}$

$F_{3} = 2 + q + 2\ q^{4} + q^{9}$

$F_{4} = 9 + 3\ q^{2} + 2\ q^{4} + 6\ q^{6} + 3\ q^{10} + q^{16}$

$F_{5} = 44 + q + 4\ q^{2} + 11\ q^{3} + 10\ q^{5} + 4\ q^{6} + 16\ q^{7} + 4\ q^{8} + 6\ q^{9} + 5\ q^{11} + 6\ q^{13} + 4\ q^{14} + 4\ q^{18} + q^{25}$

$F_{6} = 265 + 10\ q^{2} + 38\ q^{3} + 42\ q^{4} + 10\ q^{5} + 59\ q^{6} + 10\ q^{7} + 71\ q^{8} + 54\ q^{9} + 15\ q^{10} + 16\ q^{11} + 15\ q^{12} + 16\ q^{13} + 19\ q^{14} + 16\ q^{15} + 19\ q^{16} + 15\ q^{18} + 9\ q^{20} + 10\ q^{22} + 5\ q^{24} + 5\ q^{28} + q^{36}$

$F_{7} = 1854 + q + 12\ q^{2} + 116\ q^{3} + 298\ q^{4} + 162\ q^{5} + 182\ q^{6} + 287\ q^{7} + 174\ q^{8} + 268\ q^{9} + 430\ q^{10} + 152\ q^{11} + 118\ q^{12} + 40\ q^{13} + 168\ q^{14} + 85\ q^{15} + 70\ q^{16} + 134\ q^{17} + 110\ q^{18} + 35\ q^{19} + 78\ q^{20} + 35\ q^{21} + 20\ q^{22} + 50\ q^{23} + 20\ q^{24} + 49\ q^{25} + 29\ q^{27} + 15\ q^{29} + 6\ q^{30} + 14\ q^{31} + 15\ q^{33} + 6\ q^{36} + 6\ q^{40} + q^{49}$

Description

Eigenvalues of the random-to-random operator acting on a simple module.
The simple module of the symmetric group indexed by a partition

$\lambda$ has dimension equal to the number of standard tableaux of shape

$\lambda$ . Hence, the eigenvalues of any linear operator defined on this module can be indexed by standard tableaux of shape

$\lambda$ ; this statistic gives all the eigenvalues of the operator acting on the module [1].

Map

Robinson-Schensted recording tableau

Description

Sends a permutation to its Robinson-Schensted recording tableau.
The Robinson-Schensted corrspondence is a bijection between permutations of length

$n$ and pairs of standard Young tableaux of the same shape and of size

$n$ , see [1]. These two tableaux are the insertion tableau and the recording tableau.
This map sends a permutation to its corresponding recording tableau.